Sharp Reverse Hölder property for A_\infty weights on spaces of homogeneous type

Abstract

In this article we present a new proof of a sharp Reverse Hölder Inequality for $A_\infty$ weights that is valid in the context of spaces of homogeneous type. Then we derive two applications: a precise open property of Muckenhoupt classes and, as a consequence of this last result, we obtain a simple proof of a sharp weighted bound for the Hardy-Littlewood maximal function involving $A_\infty$ constants: |M|_{L^p(w)} \leq c (\frac{1}{p-1} [w]_{A_p}[σ]_{A_\infty})^{1/p}, where $1<p<\infty$, $σ=w^{\frac{1}{1-p}}$ and $c$ depends only on the doubling constant of the measure $μ$ and the geometric constant $κ$ of the quasimetric.